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Quasisemigroups and evolution equations. (English) Zbl 1102.47026

A map \(K\) from \([0,\infty)\) to bounded linear operators \(L(X)\), with \(X\) a Banach space, is called a strongly continuous quasilinear semigroup (SCQS) if \(K(t,0)\)=id, \(K(r,t+s) = K(r+t,s)K(r,t) = K(r,t)K(r+t,s)\), \(K(t,s)x_0\to K(t_0,s_0)x_0\) as \((t,s)\to (t_0,s_0)\), and \(\| K(t,s)\|\leq M(t+s)\) with continuous increasing \(M:[0,\infty]\to[1,\infty]\), \(s,t,s_0,t_0\in [0,\infty)\), \(x_0\in X\). The associated generator (family) \(A(t)\) is defined, examples are given, a special case being the evolution operator associated with \(x'(t)= A(t)x(t)\). If \(K\) is an SCQS with generator \(A(\cdot)\) strongly continuous, and \(f:[0,T]\to D\) (domain of the \(A(\cdot))\) continuously differentiable with \(\int^t_0 K(r+s,t-s)f(s)\,ds\in D\), \(0\leq t\leq T\), then \(x'(t)=A(r+t)x(t)+f(t)\), \(0<t\leq T\), \(x(0)=x_0\) \(\in D\), has a unique solution, given by \(x(t)= K(r,t)x_0+\int^t_0 K(r+s,t-s)f(s)\,ds\). If \(K\) is an SCQS with generator \(A(\cdot)\) and \(B\in L(X)\), then \(A(\cdot)+B\) is the generator of an (explicitly given) SCQS.

MSC:

47D06 One-parameter semigroups and linear evolution equations
35F10 Initial value problems for linear first-order PDEs
34G10 Linear differential equations in abstract spaces
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